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ichigo444

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what is the difference of a span of a vector and a linear combination of a vector?

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- Thread starter ichigo444
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- #1

ichigo444

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what is the difference of a span of a vector and a linear combination of a vector?

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quasar987

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- #3

annoymage

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ie: The span of the vectors v_1,...,v_n = The span of the vectors u_1,...,u_k.

- #4

quasar987

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If this is so, then span(u_1,...,u_k)=span(v_1,...,v_n). If not, then the spans are not equal.

Make sure you see why.

- #5

annoymage

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Make sure you see why.

i can't clearly see it, every element in span(u_1,...,u_k) is then the element of span(v_1,...,v_n), and conversely,

then span(u_1,...,u_k) is subset of span(v_1,...,v_n), and conversely

is that really correct?

- #6

quasar987

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[tex]u_i = \sum_{j=1}^nc_i^jv_j[/tex]

Then, for an arbitrary linear combination of the u_i,

[tex]\sum_{i=1}^ka^iu_i=\sum_{i=1}^ka^i\left(\sum_{j=1}^nc_i^jv_j\right)=\sum_{j=1}^n\left(\sum_{i=1}^ka^ic_i^j\right)v_j[/tex]

(a linear combination of the v_j !) This shows that [itex]\mathrm{span}(u_1,\ldots,u_k)\subset \mathrm{span}(v_1,\ldots,v_n)[/itex].

And in the same way, if each v_j can be written as a linear combination of the u_i, we obtain [itex]\mathrm{span}(v_1,\ldots,v_n)\subset \mathrm{span}(u_1,\ldots,u_k)[/itex].

And so in that case, [itex]\mathrm{span}(v_1,\ldots,v_n)= \mathrm{span}(u_1,\ldots,u_k)[/itex].

On the other hand, if for instance, u_i cannot be written as a linear combination of the v_j's, then [itex]\mathrm{span}(v_1,\ldots,v_n)\neq \mathrm{span}(u_1,\ldots,u_k)[/itex] since [itex]u_i\in \mathrm{span}(u_1,\ldots,u_k)[/itex] but [itex]u_i \notin\mathrm{span}(v_1,\ldots,v_n)[/itex].

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annoymage

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thanksssssssssss, i can see it now

- #8

HallsofIvy

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And please note that these are the span of awhat is the difference of a span of a vector and a linear combination of a vector?

A linear combination is single sum of scalars times vectors in the set. The span is the collection of

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